Devlin's
Angle

September 2001

Untying the Gordian Knot

One day, according to ancient Greek legend, a
poor peasant called Gordius arrived with his
wife in a public square of Phrygia in an ox
cart. As chance would have it, so the legend
continues, an oracle had previously informed the
populace that their future king would come into
town riding in a wagon. Seeing Gordius,
therefore, the people made him king. In
gratitude, Gordius dedicated his ox cart to
Zeus, tying it up with a highly intricate knot -
- the Gordian knot. Another oracle -- or maybe
the same one, the legend is not specific, but
oracles are plentiful in Greek mythology --
foretold that the person who untied the knot
would rule all of Asia.

The problem of untying the Gordian knot resisted
all attempted solutions until the year 333 B.C.,
when Alexander the Great -- not known for his
lack of ambition when it came to ruling Asia --
cut through it with a sword. "Cheat!" you might
cry. And although you might have been unwise to
have pointed it out in Alexander's presence, his
method did seem to go against the spirit of the
problem. Surely, the challenge was to solve the
puzzle solely by manipulating the knot, not by
cutting it.

But wait a minute. Alexander was no dummy. As a
former student of Aristotle, he would have been
no stranger to logical puzzles. After all, the
ancient Greek problem of squaring the circle is
easy to solve if you do not restrict yourself to
the stipulated tools of ruler and compass. Today
we know that the circle-squaring problem as
posed by the Greeks is indeed unsolvable. Using
ruler and compass you cannot construct a square
with the same area as a given circle. Perhaps
Alexander was able to see that the Gordian knot
could not be untied simply by manipulating the
rope.

If so, then the knot surely could not have had
any free ends. The two ends of the rope must
have been spliced together. This, of course,
would have made it a knot in the technical sense
of modern mathematicians.

Continuing under the assumption that many fine
minds had been stumped by the Gordian knot
problem, but no one had claimed the puzzle
was unsolvable, we may conclude that in
principle the knot could be untied, and everyone
who looked closely enough could see this fact.
In modern topological parlance, the loop of rope
must have been in the form of an unknot. Thus,
the Gordian knot was most likely constructed by
first splicing the two ends of a length of rope
to form a loop, and then "tying" the loop up
(i.e. wrapping it around itself in some way) to
disguise the fact that it was not really
knotted. And everyone was stumped until
Alexander came along and figured out that on
this occasion the sword was mightier than the
pen. (Of course, he did have a penchant for
coming to that conclusion.)

Now, when modern topologists study knots, they
assume the knots are constructed out of
perfectly flexible, perfectly stretchable,
infinitely thin string. Under those assumptions,
if the Gordian knot were really an unkotted
loop, then it would have been possible to untie
it, i.e., to manipulate it so it was in the form
of a simple loop that does not cross itself.

Thus, the only thing that could make it
absolutely necessary to resort to a sword to
untie it would be that the physical thickness of
the actual rope prevented the necessary
manipulations being carried out. In principle,
this could have been done. The rope could have
been thoroughly wetted prior to tying, then
dried rapidly in the sun after tying to make it
shrink.

This is the explanation proposed recently by
physicist Piotr Pieranski of the Poznan
University of Technology in Poland and the
biologist Andrzej Stasiak of the University of
Lausanne in Switzerland. Physicists are
interested in knots because the latest theories
of matter postulate that everything is made up
of tightly coiled (and maybe knotted) loops of
space-time, and biologists are interested in
knots because the long, string-like molecules of
DNA coil themselves up tightly to fit inside the
cell.

Pieranski and Stasiak have been studying knots
that can be constructed from real, physical
material, that has, in particular, a fixed
diameter. This restriction makes the subject
very different from the knot theory
traditionally studied by mathematicians.
Pieranski has developed a computer program,
called SONO (Shrink-On-No-Overlaps) to simulate
the manipulation of such knots.

Using this program, he showed that most ways of
trying to construct a Gordian knot will fail.
SONO eventually found a way to unravel them. But
recently he discovered a knot that worked. SONO
-- which had not been programmed to make use of
an algorithmic sword -- was unable to unravel
it. Maybe, just maybe, he had discovered the
actual structure of the Gordian knot! Here it
is:

To construct Pieranski's knot, you fold a
circular loop of rope and tie two multiple
overhand knots in it. You then pass the end
loops over the entangled domains. Then you
shrink the rope until it is tight. With this
structure, there is not enough rope to allow the
manipulations necessary to unravel it.